T.R.
SİİRT UNIVERSİTY
GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCE
SOME SPECIAL INTEGER SEQUENCES RELATED TO
BIPARTITE GRAPHS
M.Sc. THESIS
DIYAR OMAR MUSTAFA ZANGANA
(153114021)
Department of Mathematics
Supervisor: Asst. Prof. Dr. Abdulkadir KARAKAŞ
Co-Supervisor: Res. Asst. Dr. Ahmet ÖTELEŞ
August-2017
SİİRT
ACCEPTANCE AND APPROVAL OF THESIS
The thesis entitled “SOME SPECIAL INTEGER SEQUENCES RELATED TO
BIPARTITE GRAPHS” submitted by Diyar Omar Mustafa Zangana has been approved
as Master Thesis at Department of Mathematics, Graduate School of Natural and Applied
Sciences, Siirt University with unanimity/majority of votes by the committee listed below
on …/…/…….
Committee Members
Signature
Head
………..
Supervisor
………..
Member
………..
I approve the above results.
Assoc. Prof. Dr. Koray ÖZRENK
Director of the Graduate School of Natural and Applied Sciences
ACKNOWLEDGMENTS
Firstly the name of Allah we started and starting.
Before of all things, I want to thank God for helping me to finish my thesis, this
is a great opportunity to acknowledge and to thanks all persons without their support, and
help this thesis would have been impossible.
I would like to add a few heartfelt words for the people who were part of this thesis in
numerous ways.
I wish to thanks my supervisors Arş.Gör.Dr.Ahmet Öteleş and
Asst.Prof.Dr.Abdulkadir Karakaş for our indefatigable guidance, valuable suggestion,
moral support, constant encouragement and contribution of time for the successful
completion of thesis work. I am very grateful to there, for providing all the facilities
needed during the thesis development.
My special thanks to my family for indispensable support and encouragement
thing my study. Finally, I would like to thank all those who helped me directly or
indirectly.
Diyar Omar Mustafa ZANGANA
SİİRT-2017
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ... iii
TABLE OF CONTENTS ... iv
LIST OF TABLES ... vi
LIST OF FIGURES ... vii
LIST OF SYMBOLS ... viii
ÖZET ... ix
ABSTRACT ... x
1. INTRODUCTION ... 1
1.1. Structure of the Thesis ... 2
2. LITERATURE RESEARCH ... 3
3. PRELIMINARIES ... 14
3.1. Special Types of Matrices ... 14
3.1.1. Diagonal matrices ... 14
3.1.2. Triangular matrices ... 14
3.1.3. Permutation matrices ... 14
3.1.4. Circulant matrices ... 15
3.1.5. Toeplitz matrices ... 15
3.1.6. Hankel matrices ... 16
3.1.7. Hessenberg matrices ... 16
3.1.8. Tridiagonal matrices ... 17
3.2. Graph Theory ... 17
3.3. Graphs and Matrices ... 22
3.3.1. Incidence Matrix ... 22
3.3.2. Adjacency Matrix ... 23
3.3.3. Laplacian Matrix ... 24
3.4. Bipartite Graph and Perfect Matching ... 25
3.5. Permanents ... 27
3.6. Number Sequences ... 29
4. SOME SPECIAL INTEGER SEQUENCES RELATED TO BIPARTITE
GRAPHS ... 36
4.1. Jacobsthal Numbers and Associated Bipartite Graphs ... 36
4.1.1. Maple procedure ... 40
4.2. Bipartite Graphs Associated with Circulant Matrices ... 41
5. CONCLUSION ... 50
6. REFERENCES ... 51
LIST OF TABLES
Page
LIST OF FIGURES
Page
Figure 3.1. A graph with 6-vertices and 5-edges ... 18
Figure 3.2. A graph with 5-vertices and 7-edges ... 18
Figure 3.3. A graph with 5-vertices and 8-edges is called a (5; 8) graph ... 19
Figure 3.4. Directed graph ... 19
Figure 3.5. Undirected graph ... 20
Figure 3.6. Directed multiple graph ... 20
Figure 3.7. Un-directed multiple graph ... 20
Figure 3.8. Un-directed Pseudo graph ... 21
Figure 3.9. Directed Pseudo graph ... 21
Figure 3.10. A graph with 6-vertices and 5-edges ... 21
Figure 3.11. A graph with 5-vertices and 8-edges ... 22
LIST OF SYMBOLS
Symbol
Explanation
𝑮
: Graph
𝑨
𝒏= (𝒂
𝒊𝒋)
: Matrix
𝐴
𝑛of order 𝑛 with entries 𝑎
𝑖𝑗𝑨
(𝒏,𝒌)= (𝒂
𝒊𝒋)
: Matrix
𝐴
(𝑛,𝑘)of order 𝑛 with entries 𝑎
𝑖𝑗𝑎
𝑖𝑗= {
1, if − 1 ≤ |𝑗 − 𝑖| ≤ 𝑘 − 1
0, otherwise
𝑮(𝑨
𝒏)
: Graph with adjacency matrix
𝐴
𝑛𝑱(𝒏)
:
𝑛th Jacobsthal number
𝑭(𝒏)
:
𝑛th Fibonacci number
𝑳(𝒏)
:
𝑛th Lucas number
𝑨
𝒏𝒌:
𝑘th contraction of the matrix 𝐴
𝑛𝐩𝐞𝐫𝑨
𝒏: Permanent of the matrix
𝐴
𝑛ÖZET
YÜKSEK LİSANS
İKİ PARÇALI GRAFLARLA İLİŞKİLİ BAZI ÖZEL TAMSAYI DİZİLERİ
Diyar Omar Mustafa ZANGANA
Siirt Üniversitesi Fen Bilimleri Enstitüsü
Matematik Anabilim Dalı
Danışman
: Yrd. Doç. Dr. Abdulkadir KARAKAŞ
II. Danışman : Arş. Gör. Dr. Ahmet ÖTELEŞ
2017, 53 Sayfa
Bu tezde, iki parçalı komşuluk matrisleri 𝑛 × 𝑛 (0,1) − matrisler olan bazı belli tip iki parçalı grafları ele aldık. Sonra bu iki parçalı grafların mükemmel eşlemeleri sayılarının çok iyi bilinen sayı dizilerine (örneğin; Fibonacci Lucas Jacobsthal) eşit olduğunu gösterdik. Daha sonra, ele aldığımız bu graflar ve onların mükemmel eşlemeleriyle ilgili bazı örnekler verdik. Son olarak, bu iki parçalı grafların mükemmel eşlemeleri sayılarını hesaplamak için Maple 2016 prosedürleri sunduk.
Anahtar Kelimeler: Fibonacci dizisi, İki parçalı graf, Jacobsthal dizisi, Lucas dizisi, Mükemmel eşleme, Permanent.
x
ABSTRACT
M.Sc. THESIS
SOME SPECIAL INTEGER SEQUENCES RELATED TO
BIPARTITE GRAPHS
Diyar Omar Mustafa ZANGANA
The Graduate School of Natural and Applied Science of Siirt University
The Degree of Master of Science
In Mathematics
Supervisior
: Asst. Prof. Dr. Abdulkadir KARAKAŞ
Co-Supervisior : Res. Asst. Dr. Ahmet ÖTELEŞ
2017, 53 Pages
In this thesis, we consider some certain types of bipartite graphs whose bipartite adjacency matrices are the 𝑛 × 𝑛 (0,1) − matrices. Then we show that the numbers of perfect matchings (1 − factosrs) of these bipartite graphs are equal to the famous integer sequences (e.g. Fibonacci, Lucas, Jacobsthal). After that, we give some examples concerned with these graphs and their perfect matchings. Finally, we present Maple procedures in order to calculate the numbers of perfect matchings of the bipartite graphs.
Keywords: Bipartite graph, Fibonacci sequence, Jacobsthal sequence, Lucas sequence, Perfect matching, Permanent.
1. INTRODUCTION
Bipartite graph is a graph in which the vertices can be divided into two parts
such that no two vertices in the same part are joined by an edge. The investigation of
the properties of bipartite graphs was begun by König. His work was motivated by
an attempt to give a new approach to the investigation of matrices on determinants
of matrices. As a practical matter, bipartite graphs form a model of the interaction
between two di¤erent types of objects. For example; social network analysis, railway
optimization problem, marriage problem etc (Asratian et al., 1998).
A perfect matching (1-factors) of a graph is a matching (i.e., an independent
edge set) in which every vertex of the graph is incident to exactly one edge of
the matching. "The enumeration or actual construction of perfect matching of a
bipartite graph has many applications, for example, in maximal ‡ow problems and
in assignment and scheduling problems arising in operational research" (Minc, 1978).
The number of perfect matchings of bipartite graphs also plays a signi…cant role in
organic chemistry (Wheland, 1953).
Fibonacci, Lucas and Jacobsthal numbers which are respectively de…ned by
the recurrence relation
F (n) = F (n
1) + F (n
2) ;
F (0) = 0
and F (1) = 1
L (n) = L (n
1) + L (n
2) ;
L (0) = 2
and L (1) = 1
J (n) = J (n
1) + 2J (n
2) ;
J (0) = 0
and J (1) = 1,
for n
2, belong to a large family of positive integers. They have many interesting
properties and applications to almost every …eld of science and art. They continue
to contribute signi…cant innovations for investigations, and reveal the niceness of
mathematics in many …elds, particularly number theory (Koshy, 2001; Koshy, 2011;
Horadam, 1988).
"The permanent of an n
n
matrix A = (a
ij)
is de…ned by
per (A) =
X
Sn nY
i=1a
i (i)where the summation extends over all permutations
of the symmetric group S
n.
The permanent of a matrix is analogous to the determinant, where all of the signs
used in the Laplace expansion of minors are positive" (Minc, 1978). Permanents have
many applications in physics, chemistry, electrical engineering, graph theory etc.
Some of the most important applications of permanents are via graph theory. A more
di¢ cult problem with many applications is the enumeration of perfect matchings of
has been very popular problem.
1.1. Structure of the Thesis
The rest of the thesis is structured as follows.
In Chapter 2, we present a discussion of previously published work that I
did in this area in conjunction with other authors.
In Chapter 3, we give the fundamental de…nitions, structures and theorems
which are necessary to better understand the topics contained within this text.
In Chapter 4, we consider a bipartite graph. Then we show that the
num-bers of perfect matchings of this graph generate the Jacobsthal numnum-bers by the
contraction method. Finally, a Maple procedure is presented in order to compute
the numbers of perfect matchings of the graph.
In Chapter 5, we …rstly introduce two lemmas ralated to bipartite graphs
as-sociated with Fibonacci numbers. After that, we de…ne a bipartite graph asas-sociated
with n
n (0; 1)-circulant matrix whose the numbers of perfect matchings generate
the Lucas numbers. Finally, two Maple procedures are presented to compute the
numbers of perfect matchings in these graphs.
2.
LITERATURE RESEARCH
The purpose of this chapter is to further motivate the rest of the thesis
by presenting a discussion of previously published work that I did in this area in
conjunction with other authors.
Lee et al. (1997) consider "a bipartite graph G F
(n;2)= (f
i;j)
with bipartite
adjacency matrix is the n
n
tridiagonal matrix of the form
F
(n;2)=
0
B
B
B
B
B
B
B
B
B
B
@
1
1
0
0
1
1
1
. ..
..
.
0
1
. .. ... ... ...
..
.
. .. ... ... 1 0
..
.
. .. 1
1
1
0
0
1
1
1
C
C
C
C
C
C
C
C
C
C
A
;
(1)
with the entries are
f
ij=
(
1;
if jj
i
j
1;
0;
otherwise
". Then they obtain "the number of perfect matchings of G F
(n;2)is the (n + 1)st
Fibonacci number F (n + 1)". In other words,
per
F
(n;2)= F (n + 1) :
(2)
They also consider "a bipartite graph G F
(n;k)= (f
i;j)
with bipartite adjacency
matrix is the n
n (0; 1)-matrix of the form
F
(n;k)=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
1
1
0
0
1
1
1
1
0
0
0
1
1
1
1
0
0
. .. ... ... ...
. .. ... ...
..
.
. ..
. .. 0
..
.
. ..
1
. .. ... ... ...
0
0
1
1
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
(3)
with the entries are
f
ij=
(
1;
if
1
jj
i
j
k
1;
". Then they obtain "the number of perfect matchings of G F
(n;k)is g
k(n + k
1) ;
where
g
k(n)
is the nth k-Fibonacci number" (Lee et al., 1997). This time, they consider
"an another bipartite graph G (U
n= (u
i;j))
with bipartite adjacency matrix is the
n
n (0; 1)-matrix of the form
U
n=
0
B
B
B
B
B
B
B
B
B
B
@
1
1
1
1
0
1
1
0
1
0
1
1
..
.
. .. ... ... ... ...
..
.
. .. 1
0
1
0
0
1
1
1
C
C
C
C
C
C
C
C
C
C
A
with the entries are
u
ij=
8
>
<
>
:
1;
if i = j = 1 or i = j = n;
1;
if i < j or j
i =
1
0;
otherwise
". Then they obtain "the number of perfect matchings of G (U
n)
is the (n + 1)st
Fibonacci number F (n + 1)" (Lee et al., 1997).
Lee (2000) considers "a bipartite graph G L
(n;2)= (l
i;j)
with bipartite
adjacency matrix is the n
n
matrix of the form
L
(n;2)=
0
B
B
B
B
B
B
B
B
B
B
@
1
0
1
0
0
1
1
1
0
..
.
0
1
. .. ... ... 0
..
.
. .. ... ... ... 0
..
.
. .. ... ... 1
0
0
1
1
1
C
C
C
C
C
C
C
C
C
C
A
;
with the entries are
l
i;j=
8
>
>
<
>
>
:
1;
if i = j = 1 or i = 1 and j = 3;
1;
if i > 2 and jj
i
j
1;
0;
otherwise
". Then he shows that "the number of perfect matchings of G L
(n;2)is (n
1)st
Lucas number L (n
1)
for n
3
". He also considers "a bipartite graph G L
(n;k)with bipartite adjacency matrixL
(n;k)=
F
(n;k)+ E
1;k+1P
kj=2E
1;jfor n
3, where
j)-entry and zeros elsewhere". Namely,
L
(n;k)=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
0
0
1
0
1
1
1
1
0
0
0
1
1
1
1
0
0
. .. ... ... ...
. .. ... ...
..
.
. ..
. .. 0
..
.
. ..
1
. .. ... ... ...
0
0
1
1
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
Then he shows that "the number of perfect matchings of G L
(n;k)is l
k(n
1) ;
where
l
k(n)
is the nth k-Lucas number" (Lee, 2000). This time, he de…nes "the matrix
B
nas
B
n=
F
(n;2)+ E
13E
23+ E
24E
34where F
(n;2)is the matrix in (1) and E
i;jdenotes the n
n
matrix with 1 at the
(i; j)-entry and zeros elsewhere". Let G (B
n)
be the bipartite graph with bipartite
adjacency matrix B
n:
Then he shows that "the number of perfect matchings of
G (B
n)
is (n
1)st Lucas number L (n
1)" (Lee, 2000).
Shiu et al. (2003) …rstly de…ne the (k; )-sequences s
k(n) :
Then they give
the following result:
"For a …xed m
1, suppose n; k
2
and n
m:
Let G B
(n;k)a bipartite
graph with bipartite adjacency matrix has the form
B
(n;k)=
0
B
B
B
B
B
B
B
@
a
1a
2: : :
a
m0 : : : 0
1
0
F
(n 1;2)..
.
0
1
C
C
C
C
C
C
C
A
;
for some elements a
1; a
2; : : : ; a
min a ring R:Then the number of perfect matching
of G B
(n;k)is nth (k; )-number s
k(n)
with
= (a
1; a
2; : : : ; a
m)".
bi-partite adjacency matrix has the form
V
n=
0
B
B
B
B
B
B
B
B
B
B
B
B
@
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
0
0
..
.
. .. ... ... ... ... ...
..
.
. .. ... ... ... 0
..
.
. .. ... ... 1
0
0
1
1
1
C
C
C
C
C
C
C
C
C
C
C
C
A
with the entries are
v
ij=
(
1;
if
1
j
i
1
or i = 1;
0;
otherwise
". Then they obtain that "the number of perfect matchings of G (V
n)
is
P
ni=0
F (i) =
F (n + 2)
1;
where F (n) is the nth Fibonacci number". They also consider "a
bipartite graph G (W
n)
with bipartite adjacency matrix W
n= V
n+ Y
n;
where Y
nde-notes the n
n
matrix with
1
at the (1; 2)-entry, 1 at the (2; 4)-entry and zeros
elsewhere". Clearly,
W
n=
0
B
B
B
B
B
B
B
B
B
B
B
B
@
1
0
1
1
1
1
1
1
1
0
0
0
1
1
1
0
0
..
.
. .. ... ... ... ... ...
..
.
. .. ... ... ... 0
..
.
. .. ... ... 1
0
0
1
1
1
C
C
C
C
C
C
C
C
C
C
C
C
A
Then for n
4;
they obtain that "the number of perfect matchings of G (W
n)
is
P
n 2i=0
L (i) = L (n)
1;
where L (n) is the nth Lucas number" (K¬l¬ç and Ta¸
sc¬,
2008a).
K¬l¬ç and Ta¸
sc¬(2008b) consider "a bipartite graph G M
(n;k)with bipartite
adjacency matrix M
(n;k)=
F
(n;k)+U
(n;k);
where F
(n;k)is the matrix given by (3) and
0": Clearly,
M
(n;k)=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
0
1
1
1
1
0
0
0
..
.
. .. ... ... ... ... ... ... ... ...
..
.
0
1
1
1
1
0
0
..
.
0
1
1
1
1
0
0
..
.
0
1
1
1
1
0
0
0
1
1
1
1
0
0
1
1
1
0
0
1
1
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
:
(4)
Then for n
3;
they obtain that "the number of perfect matchings of G M
(n;k)is
the nth k-Lucas number, l
k(n)". They also consider "a bipartite graph G T
(n;k)with bipartite adjacency matrix T
(n;k)=
F
(n;k)+ V
(n;k)for 2
k < n;
where F
(n;k)is the matrix given by (3) and V
(n;k)= (v
ij)
is the n
n (0; 1)-matrix with v
1j= 1
if k + 1
j
n
and otherwise 0": Clearly,
T
(n;k)=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
1
1
1
1
0
0
..
.
. .. ... ... ... ... ... ... ...
..
.
..
.
0
1
1
1
1
. .. ...
..
.
0
1
1
1
1
0
..
.
0
1
1
1
1
0
0
1
1
1
0
0
1
1
1
0
0
1
1
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
:
Then they obtain that "the number of perfect matchings of G T
(n;k)is the sums
of k-Fibonacci numbers,
P
nj=1g
k(j)
for n
3" (K¬l¬ç and Ta¸
sc¬, 2008b). This
time, they consider "a bipartite graph G E
(n;k)with bipartite adjacency matrix
E
(n;k)= M
(n;k)+ D
(n;k)for 2
k < n;
where M
(n;k)is the matrix given by (4) and
0": Clearly,
E
(n;k)=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
1
1
1
1
0
0
0
0
..
.
. .. ... ... ... ... ... ... ... ... ...
..
.
0
1
1
1
1
0
0
..
.
0
1
1
1
1
0
0
..
.
0
1
1
1
1
0
0
0
1
1
1
1
0
0
1
1
1
0
0
1
1
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
:
Then they obtain that "the number of perfect matchings of G E
(n;k)is the sums
of k-Lucas numbers,
P
n 1j=1l
k(j)
for n
3" (K¬l¬ç and Ta¸
sc¬, 2008b).
K¬l¬ç and Stakhov (2009) consider "a bipartite graph G M
(n;p)with
bipar-tite adjacency matrix M
(n;p)= (m
ij)
with m
i+1;i= m
i;i= m
i;i+p= 1
for a …xed
integer p and otherwise 0": Clearly,
M
(n;p)=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
0
0
1
0
0
1
1
0
0
1
. .. ...
0
1
1
0
0
1
0
0
0
1
1
0
0
1
..
.
. .. ... ... ... ...
0
..
.
. .. ... ... ... ... ...
..
.
. .. ... ... ... 0
0
0
0
1
1
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
:
(5)
Then they obtain that "the number of perfect matchings of G M
(n;p)is the (n + 1)st
generalized Fibonacci p-number F
p(n + 1)
for n
3". They also consider a bipartite
graph G T
(n;p)with bipartite adjacency matrix has the form
T
(n;p)=
0
B
B
B
B
B
B
B
@
1 1 1
1 1
1
0
M
(n 1;p)..
.
0
1
C
C
C
C
C
C
C
A
;
where M
(n 1;p)is the matrix of order (n
1)
given by (5)". Then they obtain
that "the number of perfect matchings of G T
(n;p)is the sums of the consecutive
generalized Fibonacci p-numbers,
P
ni=1F
p(i)
for n
3" (K¬l¬ç and Stakhov, 2009).
Akbulak and Ötele¸
s (2013) give the following results for the number of
per-fect matchings in bipartite graphs with upper Hessenberg adjacency matrix related
to Fibonacci, Lucas and Padovan numbers.
"Let G(U
n= (u
ij))
be a bipartite graph with bipartite adjacency matrix is
the n
n (0; 1)-upper Hessenberg matrix de…ned by
U
n=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
0
1
0
1
0
1
1
1
0
1
0
1
0
1
0
1
1
0
1
0
1
0
1
0
0
1
1
0
1
0
1
0
1
0
0
1
1
0
1
0
1
0
0
0
1
1
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
(6)
where
u
ij=
8
>
<
>
:
1;
if j
i =
1;
1;
if i
j
and j
i
0 (mod 2) ;
0;
otherwise.
Then the number of perfect matchings of G(U
n)
is the nth Fibonacci number F (n)
for n
3" (Akbulak and Ötele¸
s, 2013).
"Let G(V
n= (v
ij))
be a bipartite graph with bipartite adjacency matrix is
the n
n (0; 1)-upper Hessenberg matrix de…ned by
V
n=
0
B
B
B
B
B
B
B
B
B
@
1 1 1
1
1 0 1
0
1
0 1
0 0
U
n 2..
.
..
.
0 0
1
C
C
C
C
C
C
C
C
C
A
;
where U
n 2is the matrix of order (n
2)
in 6. Then the number of perfect matchings
of G(V
n)
is the (n
1)st Lucas number L (n
1)
for n
3" (Akbulak and Ötele¸
s,
2013).
the n
n (0; 1)-upper Hessenberg matrix de…ned by
W
n=
0
B
B
B
B
B
B
B
B
B
B
B
@
1
1
1
1
1
1
1
1
0
1
0
0
1
0
0
1
0
1
0
0
1
0
0
0
1
0
1
0
0
1
0
0
0
1
0
1
C
C
C
C
C
C
C
C
C
C
C
A
;
where
w
ij=
8
>
<
>
:
1;
if i = 1 or j
i =
1;
1;
if i
j
and j
i
1 (mod 3) ;
0;
otherwise.
Then the number of perfect matchings of G(W
n)
is the nth Padovan number P (n)
for n
3" (Akbulak and Ötele¸
s, 2013).
Ötele¸
s (2017) gives the following results for the number of perfect matchings
in bipartite graphs related to the well-known integer sequences (Fibonacci, Lucas
i.e.).
"Let G (H
n= (h
i;j)) (n = 2t; t
2 N) be a bipartite graph with bipartite
adjacency matrix is the n
n (0; 1)-tridiagonal matrix has the form
H
n=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
1
0
0
1
0
1
. ..
..
.
0
1
1
1
. ..
..
.
..
.
. .. 1
0
1
. .. ...
..
.
. .. 1 ... 1 0
..
.
. .. ...
1
0
0
1
1
C
C
C
C
C
C
C
C
C
C
C
C
C
A
;
(7)
where
h
i;j=
8
>
>
<
>
>
:
1;
if jj
i
j = 1;
1;
if i = j = 2m
1 (m
2 N) ;
0;
otherwise.
Then the number of perfect matchings of G (H
n)
is 1" (Ötele¸
s, 2017).
"Let G (H
n) (n = 2t + 1; t
2 N) be a bipartite graph with bipartite
adja-cency matrix H
ngiven by (7). Then the number of perfect matchings of G (H
n)
is
n+1
2
" (Ötele¸
s, 2017).
the n
n (0; 1)-tridiagonal matrix has the form
K
n=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
1
0
0
1
0
1
. ..
..
.
0
1
1
1
. ..
..
.
..
.
. .. 1
0
1
. .. ...
..
.
. .. 1
1
1
0
..
.
. .. ... ... 1
0
0
1
1
1
C
C
C
C
C
C
C
C
C
C
C
C
C
A
;
where k
2;2= k
4;4= 0;
all other terms on the main diagonal are 1; all terms on the
sub-diagonal and superdiagonal are 1 and otherwise k
i;j= 0. Then for n
3, the
number of perfect matchings of G (K
n)
is the (n
3)rd Lucas number L (n
3)"
(Ötele¸
s, 2017).
"Let G (B
n= (b
i;j))
be a bipartite graph with bipartite adjacency matrix is
the n
n (0; 1)-anti-tridiagonal matrix has the form
B
n=
0
B
B
B
B
B
B
B
B
B
B
@
0
0
1
1
..
.
. .
.
1
1
1
..
.
. .
.
. .
.
. .
.
1
0
0
. .
.
. .
.
. .
.
. .
.
..
.
1
1
. .
.
. .
.
..
.
1
1
0
0
1
C
C
C
C
C
C
C
C
C
C
A
;
where
b
i;j=
(
1;
if ji + j mod (n + 1)j
1;
0;
otherwise.
Then for n
2, the number of perfect matchings of G (B
n)
is the (n + 1)st Fibonacci
number F (n + 1)" (Ötele¸
s, 2017).
"Let G (D
n= (d
i;j))
be a bipartite graph with bipartite adjacency matrix
has the form
D
n=
0
B
B
B
B
B
B
B
B
B
B
@
0
0
1
0
1
..
.
0
1
1
1
..
.
. .
.
. .
.
. .
.
1
0
0
. .
.
. .
.
. .
.
. .
.
..
.
1
1
1
. .
.
..
.
1
1
0
0
1
C
C
C
C
C
C
C
C
C
C
A
where
d
i;j=
8
>
>
<
>
>
:
1;
if i = 1 and j = n
2; j = n;
1;
if ji + j mod (n + 1)j
2 i n1;
0;
otherwise.
Then for n
3, the number of perfect matchings of G (D
n)
is the (n
1)st Lucas
number L (n
1)" (Ötele¸
s, 2017).
"Let G (U
n= (u
i;j))
be a bipartite graph with bipartite adjacency matrix
has the form
U
n=
0
B
B
B
B
B
B
B
B
B
B
B
B
@
1
1
1
1
1
0
0
1
1
1
..
.
. .
.
1
1
1
0
..
.
. .
.
. .
.
. .
.
1
. .
.
0
1
. .
.
. .
.
. .
.
..
.
1
1
1
. .
.
0
1
1
0
0
0
1
C
C
C
C
C
C
C
C
C
C
C
C
A
where
u
i;j=
8
>
>
<
>
>
:
1;
if i = 1 and 1
j
n;
1;
if ji + j mod (n + 1)j
2 i n1;
0;
otherwise.
Then for n
2, the number of perfect matchings of G (U
n)
is F (n + 2)
1, where
F (n)
is the nth Fibonacci number" (Ötele¸
s, 2017).
"Let G (V
n= (v
i;j))
be a bipartite graph with bipartite adjacency matrix
has the form
V
n=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
1
1
1
0
1
0
0
1
1
1
1
..
.
. .
.
0
1
1
1
0
..
.
. .
.
. .
.
. .
.
1
1
0
0
0
. .
.
. .
.
. .
.
1
. .
.
. .
.
..
.
0
1
1
. .
.
. .
.
. .
.
..
.
1
1
1
0
0
0
1
1
0
0
0
0
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
where
v
i;j=
8
>
>
>
>
>
<
>
>
>
>
>
:
1;
if i = 1 and 1
j
n
2; j = n;
1;
if i = 2 and j = n
3;
1;
if ji + j mod (n + 1)j
2 i n1;
0;
otherwise.
is the nth Lucas number" (Ötele¸
s, 2017).
"Let G (P
n= (p
ij))
be a bipartite graph with bipartite adjacency matrix is
the n
n (0; 1)-pentadiagonal matrix has the form
P
n=
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
0
1
0
0
0
1
0
1
. ..
..
.
1
0
1
. .. ... ... ...
0
1
. .. ... ... 1 0
..
.
. .. ... ... ... 0 1
..
.
. .. 1
0
1
0
0
0
1
0
1
1
C
C
C
C
C
C
C
C
C
C
C
C
C
A
:
Then the number of perfect matchings of G (P
n)
is obtained as
per (P
n) =
8
<
:
F
n+12F
n+32;
if n is odd,
F
n2+ 1
2;
if n is even,
3. PRELIMINARIES
In this chapter, we give the fundamental de…nitions, structures and theorems
which are necessary to better understand the topics contained within this text.
3.1. Special Types of Matrices
3.1.1. Diagonal matrices
"A matrix D = [d
ij]
2 M
n;m(F )
is diagonal if d
ij= 0
whenever j 6= i" (Horn
and Johnson, 1990).
3.1.2. Triangular matrices
"A matrix T = [t
ij]
2 M
n;m(F )
is upper triangular if t
ij= 0
whenever
i > j. If t
ij= 0
whenever i
j
, then T is said to be strictly upper triangular.
Analogously, T is lower triangular (or strictly lower triangular) if its transpose is
upper triangular (or strictly upper triangular). A triangular matrix is either lower
or upper triangular; a strictly triangular matrix is either strictly upper triangular
or strictly lower triangular" (Horn and Johnson, 1990).
3.1.3. Permutation matrices
"A square matrix P is a permutation matrix if exactly one entry in each
row and column is equal to 1 and all other entries are 0. Multiplication by such
matrices e¤ects a permutation of the rows or columns of the matrix multiplied. For
example,
0
B
@
0 1 0
1 0 0
0 0 1
1
C
A
0
B
@
1
2
3
1
C
A =
0
B
@
2
1
3
1
C
A
illustrates how a permutation matrix produces a permutation of the rows (entries)
of a vector: it sends the …rst entry to the second position, sends the second entry
to the …rst position, and leaves the third entry in the third position" (Horn and
Johnson, 1990).
3.1.4. Circulant matrices
"A matrix A 2 M
n(F )
of the form
A =
0
B
B
B
B
B
B
B
@
a
1a
2a
na
na
1a
2a
n 1a
n 1a
na
1a
n 2..
.
..
.
. .. ...
..
.
a
2a
3a
na
11
C
C
C
C
C
C
C
A
is a circulant matrix. Each row is the previous row cycled forward one step; the
en-tries in each row are a cyclic permutation of those in the …rst. The n n permutation
matrix
C =
0
B
B
B
B
B
B
B
@
0 1
0
0
..
.
0
1
..
.
. .. ... 0
0
1
1 0
0
1
C
C
C
C
C
C
C
A
=
0
I
n 11 0
1;n 1!
is the basic circulant permutation matrix. A matrix A 2 M
n(F )
can be written in
the form
A =
n 1X
k=0a
k+1C
nk(a polynomial in the matrix C
n) if and only if it is a circulant. We have C
n0= I =
C
nn
;
and the coe¢ cients a
1; : : : ; a
nare the entries of the …rst row of A" (Horn and
Johnson, 1990).
3.1.5. Toeplitz matrices
"A matrix A = a
ij2 M
n+1(F )
of the form
A =
0
B
B
B
B
B
B
B
B
B
B
@
a
0a
1a
2a
na
1a
0a
1a
2a
n 1a
2a
1a
0a
1a
n 2..
.
..
.
. .. ... ...
..
.
..
.
..
.
. .. ...
a
1a
na
n+1a
1a
01
C
C
C
C
C
C
C
C
C
C
A
is a Toeplitz matrix.
The entry a
ijis equal to a
j ifor some given sequence
the diagonals parallel to the main diagonal. The Toeplitz matrices
B =
0
B
B
B
B
B
@
0 1
0
0
. ..
. .. 1
0
0
1
C
C
C
C
C
A
and
F =
0
B
B
B
B
@
0
0
1
0
. .. ...
0
1
0
1
C
C
C
C
A
are called the backward shift and forward shift because of their e¤ect on the elements
of the standard basis fe
1; : : : ; e
n+1g. Moreover, F = B
Tand B = F
T. A matrix
A
2 M
n+1can be written in the form
A =
nX
k=1a
kF
k+
nX
k=0a
kB
kif and only if it is a Toeplitz matrix. Toeplitz matrices arise naturally in problems
involving trigonometric moments" (Horn and Johnson, 1990).
3.1.6. Hankel matrices
"A matrix A 2 M
n+1(F )
of the form
A =
0
B
B
B
B
B
B
B
@
a
0a
1a
2a
na
1a
2. .
.
a
n+1a
2. .
.
..
.
..
.
a
na
2n 1a
na
n+1a
2n 1a
2n1
C
C
C
C
C
C
C
A
is a Hankel matrix . Each entry a
ijis equal to a
i+j 2for some given sequence
a
0; a
1; a
2; : : : ; a
2n 1; a
2n:
The entries of A are constant along the diagonals
perpen-dicular to the main diagonal. Hankel matrices arise naturally in problems involving
power moments" (Horn and Johnson, 1990).
3.1.7. Hessenberg matrices
be an upper Hessenberg matrix if a
ij= 0
for all i > j + 1 :
A =
0
B
B
B
B
B
B
B
@
a
11F
a
21a
22a
32. ..
. ..
. ..
0
a
n;n 1a
nn1
C
C
C
C
C
C
C
A
An upper Hessenberg matrix A is said to be unreduced if all its sub-diagonal entries
are nonzero, that is, if a
i+1; i = 0
for all i = 1; : : : ; n
1;
the rank of such a matrix
is at least n
1
since its …rst n
1
columns are independent" (Horn and Johnson,
1990).
3.1.8. Tridiagonal matrices
"A matrix A = [a
ij]
2 M
n(F )
that is both upper and lower Hessenberg is
called tridiagonal, that is, A is tridiagonal if a
ij= 0
whenever ji
j
j > 1:
A =
0
B
B
B
B
B
@
a
1b
10
c
1a
2. ..
. .. ... b
n 10
c
n 1a
n1
C
C
C
C
C
A
" (Horn and Johnson, 1990).
3.2. Graph Theory
De…nition 1
"A graph G consists of a set of objects V = fv
1; v
2; v
3; : : :
g called
ver-tices (also called points or nodes) and other set E = fe
1; e
2; e
3; : : :
g whose elements
are called edges (also called lines or arcs)" (Vasudev, 2006).
De…nition 2
"The set V (G) is called the vertex set of G and E(G) is the edge set.
Usually the graph is denoted as G = (V; E)" (Vasudev, 2006).
Let G be a graph and fu; vg an edge of G. Since fu; vg is 2-element set, we may
write fv; ug instead of fu; vg. It is often more convenient to represent this edge by
uv
or vu: If e = uv is an edge of a graph G, then we say that u and v are adjacent in
G
and that e joins u and v. (We may also say that each that of u and v is adjacent
to or with the other).
and
E(G) =
fuv; uw; wx; xy; xzg:
Now we have the graph in the Figure 3.1 by considering these sets.
Figure 3.1. A graph with
6
-vertices and5
-edgesEvery graph has a diagram associated with it. The vertex u and an edge e
are incident with each other as are v and e. If two distinct edges say e and f are
incident with a common vertex, then they are adjacent edges.
Example 4
In Figure 3.2 the vertices a and b are adjacent but a and c are not.
The edges x and y are adjacent but x and z are not. Although the edges x and z
intersect in the diagram, their intersection is not a vertex of the graph.
Figure 3.2. A graph with
5
-vertices and7
-edgesDe…nition 5
"A graph with p-vertices and q-edges is called a (p; q) graph. The
(1; 0) graph is called trivial graph" (Vasudev, 2006).
Example 6
Let V = f1; 2; 3; 4g and E = ff1; 2g; f1; 3g; f3; 2g; f4; 4gg:Then G(V; E)
is a graph.
Example 7
Let V = f1; 2; 3; 4g and E = ff1; 5g; f2; 3gg:Then G(V; E) is not a
Figure 3.3. A graph with
5
-vertices and8
-edges is called a(5; 8)
graphDe…nition 8
"A directed graph or digraph G consists of a set V of vertices and a
set E of edges such that e 2 E is associated with an ordered pair of vertices. In other
words, if each edge of the graph G has a direction then the graph is called directed
graph" (Vasudev, 2006).
In the diagram of directed graph, each edge e = (u; v) is represented by an
arrow or directed curve from initial point u of e to the terminal point v. Figure 3.4
is an example of a directed graph.
Figure 3.4. Directed graph
Suppose e = (u; v) is a directed edge in a digraph, then
(i) u is called the initial vertex of e and v is the terminal vertex of e
(ii) e is said to be incident from u and to be incident to v.
(iii) u is adjacent to v, and v is adjacent from u.
De…nition 9
"An un-directed graph G consists of set V of vertices and a set E of
edges such that each edge e 2 E is associated with an unordered pair of vertices. In
other words, if each edge of the graph G has no direction then the graph is called
We
can refer to an edge joining the vertex pair i and j as either (i; j) or
(j;
i): Figure 3.5 is an example of an undirected graph.
Figure 3.5. Undirected graph
De…nition 10
"An edge of a graph that joins a node to itself is called loop or self
loop" (Vasudev, 2006).
For example, a loop is an edge (v
i; v
j)
where v
i= v
j.
De…nition 11
"A multi-graph is a graph which is permitted to have multiple edges
(also called parallel edges) that is, edges that have the same end nodes" (Vasudev,
2006).
Two edges (v
i; v
j)
and (v
f; v
r)
are parallel edges if v
i= v
fand v
j= v
r.
Figure 3.6. Directed multiple graph Figure 3.7. Un-directed multiple graph
In Figure 3.6, there are two parallel edges associated with v
2and v
3. In Figure 3.7,
there are two parallel edges joining nodes v
1and v
2and v
2and v
3(Vasudev, 2006).
De…nition 12
"A graph in which loops and multiple edges are allowed, is called a
pseudo graph.
Figure 3.8. Un-directed Pseudo graph Figure 3.9. Directed Pseudo graph
". (Vasudev, 2006).
De…nition 13
"A graph which has neither loops nor multiple edges. i.e., where
each edge connects two distinct vertices and no two edges connect the same pair of
vertices is called a simple graph" (Vasudev, 2006).
Figure 3.4 and 3.5 represents simple un-directed and directed graph because
the graphs do not contain loops and the edges are all distinct.
De…nition 14
"A graph with …nite number of vertices as well as a …nite number of
edges is called a …nite graph. Otherwise, it is an in…nite graph" (Vasudev, 2006).
De…nition 15
"The number of edges incident on a vertex v
iwith self-loops counted
twice (is called the degree of a vertex v
iand is denoted by deg
G(v
i) or degv
ior d(v
i)"
(Vasudev, 2006).
The degrees of vertices in the graph G and H are shown in Figure 3.10 and
3.11.
and
Figure 3.11. A graph with